Abstract
We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a ``reference metric" which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaître-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reach our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.
Highlights
The last 15 years have seen several developments about massive gravity originating in the DGP model [1] and its ability to produce a late time acceleration of the Universe via a large distance modification of gravity [2, 3]
A family of massive gravity theories was recently proposed by de Rham, Gabadadze, and Tolley [17,18,19] in which the absence of ghost was first addressed in the so-called decoupling limit [18] and fully shown at the nonlinear level1 by a Hamiltonian analysis, later extended to bimetric theories [24,25,26,27]
In this paper we have given the detailed derivation of the field equations for a massive graviton propagating on an arbitrary background metric as obtained from the de Rham-Gabadadze-Tolley (dRGT) model
Summary
The last 15 years have seen several developments about massive gravity originating in the DGP model [1] and its ability to produce a late time acceleration of the Universe via a large distance modification of gravity [2, 3]. The unique consistent theory for a free (non self interacting) massive spin-2 field on a Minkowski space-time has been known for a long time as the Fierz-Pauli theory [7] It propagates 5 degrees of freedom of positive energy, those of a transverse, traceless, symmetric, two times covariant tensor. One of these d.o.f., a scalar mode, leads to the well known van-Dam Veltman Zakharov discontinuity: namely the fact that, small the graviton mass, Fierz-Pauli theory leads to different physical predictions (such as light bending) from those of linearized General Relativity [8].
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